Title: On empirical Hodge Laplacians under the manifold hypothesis Show Chinese title

Title: 关于流形假设下的经验霍奇拉普拉斯算子

Abstract: Given i.i.d. observations uniformly distributed on a closed submanifold of the Euclidean space, we study higher-order generalizations of graph Laplacians, so-called Hodge Laplacians on graphs, as approximations of the Laplace-Beltrami operator on differential forms. Our main result is a high-probability error bound for the associated Dirichlet forms. This bound improves existing Dirichlet form error bounds for graph Laplacians in the context of Laplacian Eigenmaps, and it provides insights into the Betti numbers studied in topological data analysis and the complementing positive part of the spectrum. Show Chinese abstract

Abstract: 给定欧几里得空间中均匀分布在闭子流形上的独立同分布观察样本，我们研究图拉普拉斯算子的高阶推广，即所谓的图上的霍奇拉普拉斯算子，将其作为微分形式上的拉普拉斯-贝尔特拉米算子的逼近。我们的主要结果是与之相关的狄利克雷型在高概率下的误差界。这一界改进了拉普拉斯特征映射背景下图拉普拉斯算子现有的狄利克雷型误差界，并且为拓扑数据分析中研究的贝蒂数以及谱的补充正部提供了见解。